We have a functions manager who uses MSOutlook on a PC and sends an
invite out for appointments for all of her functions. The recipients
accept the invite and the tracking (in the appointment) shows the
recipient has accepted. The appointment is then in the Function
Manager's calendar as well as the recipient's calendar. Any updates are
then sent and the appointment is also updated in the recipient's calendar.

That is the theory part. The problem is, a number of people who work
there are now using their own Mac laptops and iPads. For some reason, it
does not work (as listed above) on the Mac devices. Even though the Mac
recipients (not using Mac Outlook but the included Mail program) respond
to the invite, it is not showing as accepted in the PC Outlook. The
updating is the important part as this refers to times, number of people
attending and resources required. This is all needed for rostering, etc.

Have you verified how the people using Macs are responding to the invite?

My experience has been, frustratingly, that these invite requests show up as a blank e-mail in Mail.app with an attachment. You must open the attachment with iCal and accept the invitation there.